Remark 1.2 Rational points on non-singular quadric surfaces are known to conform to the classical Manin's conjecture [1]}. Let \(A = a_1\cdots a_4\) . It can be shown that the Picard group of the quadric \(F_{\mathbf {a}}(\mathbf {x})=0\)  has rank 1 when \(A\ne \square \)  and rank 2 when \(A=\square \) . Thus Manin's conjecture predicts that
\(N_{\mathbf {a}}(B)\sim {\left\lbrace \begin{array}{ll}c_{\mathbf {a}}B, &\textrm { if }A \ne \square \\c^{\prime }_{\mathbf {a}}B\log B, &\textrm { if }A=\square ,\end{array}\right.}\) 
